Another categorial update

It’s been a month since I last posted about the category theory project, so a quick update  — and the end really is in sight!

  • I’ve just put online another revised version of Category Theory I. Little has changed except for some more corrections of typos (with particular thanks to Georg Meyer) and a few small changes for added clarity (with particular thanks to John Zajac). I’ve also made a few very small changes to better fit with what happens in Category Theory II as I steadily revise that.
  • More significantly, there is another version of Category Theory II linked on the category theory page. The old chapters on adjunctions are now in a much better state. I don’t think I found any horrendous errors, but the story is (I certainly hope!) a lot clearer in a number of key places.
  • In fact, the bit of recent work on this that I’m most pleased with is probably the proof of ‘RAPL’ (Right Adoints Preserve Limits). Tom Leinster and Steve Awodey offer fancy-but-unilluminating proofs. I spell out the sort of bread-and-butter proof idea I got from Peter Johnstone’s lectures (and my version is perhaps a little clearer for a first encounter than Emily Riehl’s?).
  • It’s a judgement call where to stop. For example, I still reckon (as I did before) that the Adjoint Functor Theorems are just over the boundary, as far as what is really appropriate for an entry-level introduction. However, I do now say just a very little about monads (so at least you know what the idea is), though I might yet add another example or two.
  • I still need to revise the last three chapters of Category Theory II. They should be in a reasonable basic state as these are the same final chapters that — in the previous arrangement — appeared as the last chapters in the 2023 published version of Category Theory I. However, in the somewhat more advanced context of Category Theory II it might be appropriate to expand the discussions a bit.

I’d hoped that Category Theory II would be paperback-ready by the end of this month. There have been unforeseen distractions. But I’m not far off. Watch this space.

More Cambridge Elements: Mark Wilson on maths, David Liggins on abstract objects

I overlooked Mark Wilson’s Innovation and Certainty when it was published in 2020. But I didn’t miss much. The topic is an excellent one — just what is going on when we make innovations like adding points and lines at infinity in geometry (to take a reasonably comfortable but still instructive example), and just how are these extensions justified? How can we be sure they don’t lead us astray? But heavens, Wilson’s discussion is arm-wavingly pretentious and tediously obscurantist. It is just a dreadful piece of writing, and it baffles me that the series editor let it pass. Don’t waste time on this.

By contrast, David Liggins’s brand new mini-book on Abstract Objects  is the very model for how Elements should surely be.  It is admirably lucid and plain-speaking, approachable by an undergraduate student, yet the way Liggins organizes the material (evidently reflecting a good deal of thought) ought to be useful too for readers with rather more background who, for example, want to revisit the area and perhaps return to thinking about it.

I do have quibbles. Well, more than quibbles. I suspect that we have pretty different views on Hale/Wright abstractionism (which is touched on), and on the value of the ideas in e.g. Charles Parsons’s Mathematical Thought and Its Objects or Øystein Linnebo’s Thin Objects, (neither of which is mentioned). And if I weren’t telling myself that I must concentrate on other projects, I’d certainly be moved to engage properly here. But disagreement is only to be expected with a fifty-page essay. And I’d still happily put this into the hands of a student. Wilson’s effort, not so much.

(Abstract Objects is still free to download from CUP for another 48 hours or so.)

Two new Cambridge Elements on Phil. Maths

Just briefly to note that there are two new short contributions in the Cambridge Elements series in the Philosophy of Mathematics, both free to download for another week. The Euclidean Programme by Alex Paseau and Wesley Wrigley critically examines the traditional idea that mathematical knowledge is obtained by deduction from self-evident axioms or first principles. How much of that idea can be rescued?

And Number Concepts by Richard Samuels and Eric Snyder takes an interdisciplinary approach to reviewing and critically assessing work on number concepts in developmental psychology and cognitive science. (And after all, shouldn’t philosophers of arithmetic be interested in the concepts deployed by folk arithmeticians?)

So far, contributions in this series have been, it seems to me, a rather mixed bunch. So naive induction is little guide, in this case, as to how worthwhile these new efforts will prove to be. But let’s live in hope. When I’ve had a chance to take a proper look, I’ll let you know what I think. But I thought I would post a quick note straight away, while these two Elements are still freely downloadable, and you can judge them for yourself.


These days I rarely visit the philosophy news website Daily Nous. But my eye was caught by a recent post (or in fact, a re-post) inviting readers to report markedly good experiences with journals — to counterbalance the frequent complaints about various  journals for the slowness of getting any decision, and worse.  And in the replies, the journal Ergo comes in for a lot of praise as outstanding in handling submissions. That had to be a surprise to me, because (ok, I’m obviously way off the pace here!) the journal had previously never crossed my radar.

So I took a look, here. And at one level I’m hugely impressed. It’s a genuinely open-access journal; its procedures seem quite exemplary in principle, and by all accounts work excellently well in practice. The online reading experience is terrific, with a well-designed look’n’feel. And if you download a PDF of an article, it is also very decently designed. (Someone with a good eye was involved in tweaking the under-the-bonnet engines driving the site.) As you will probably know, I’m all for open-access, and Ergo seems a splendid model for journals. All credit to those involved.


But ….

When I looked at the abstracts of the forty pieces published last year how many did I actually want to read?

Pretty much zero. I did try dipping into a few on topics that I could perhaps have mustered some interest in but (no names, no pack drill) I found them laboured and unexciting, and I just wasn’t drawn in at all. What did I overlook?

Now I’m well aware that this could indeed reflect much more on my increasing distance from the fray than on the quality of the papers. But equally, I really had little sense that I was missing out on a scene of bubbling intellectual ferment. I’m almost tempted to add: not like the good old days, eh?

(Oh, and I did notice that Analysis still comes in for praise in the Daily Nous comments. That’s good to hear.)

A categorial update

Some categorial news:

  • I have just withdrawn Category Theory I: Notes towards a gentle introduction from sale as a pbk. There is going to be a new pbk edition, with a slightly different title, shortly — and now that plans are firmly under way, I don’t want anyone splashing out their hard-won pennies today only to find that a shiny new update is available a few weeks later.
  • You can download a draft of the new Category Theory I: A gentle prologue from the category theory page here. The obvious major change is that the chapters on elementary toposes at the end of the previous version have been moved to Part II, and a few initial chapters on functors moved from Part II into Part I. One result is that all of Category Theory I can indeed be thought of as a gentle prologue to some core topics in category theory.
  • The current draft of Category Theory II can also be downloaded. It has the new subtitle Four basic themes with groups of chapters covering (A) more on functors, including natural transformations, (B) around and about Yoneda, (C) adjunctions (D) a little on elementary toposes. I hope that a pbk version will be done and dusted by Easter.
  • I would still hugely welcome comments and corrections on both Parts. And indeed, even when paperbacked for those who like me prefer working from a printed copy, I’ll continue to think of them as beta versions — largely functional and I hope not too buggy but still work in progress.

It’s been sort-of enjoyable trying to get this stuff straighter in my mind, and a few friendly souls have told me that they’ve found my efforts helpful. But it really is (past) time I got back to other logic matters ….

Schubert on Sunday 8: Julian Prégardien and Els Biesemans, Die Schöne Müllerin

Julian Prégardien, in wonderful voice, is utterly compelling. The plangent tones of the fortepiano and Els Biesemans’ utter involvement adds so much. The shared level of commitment makes for a heartbreaking performance, of great emotional intensity. Surely one of the very finest recorded performances on disc or otherwise that we have.

Why it is in grim times that such wrenching music can yet be, in its way, consoling I cannot tell.

Beaney’s Tractatus translation

I’m not sure what prompted me to send off for a copy of the new translation of the Tractatus by Michael Beaney (it is, though, a very inexpensive paperback from Oxford World Classics, for which OUP are to be thanked).

A quick description. The initial apparatus is almost a hundred pages. There is a sixty page Introduction, very much aimed at the new student reader. Then an eighteen page Note on the Text which goes into probably unnecessary detail. There is a daunting Bibliography and a short Wittgenstein Chronology. Then comes the Translation — but no German text on facing pages on the rather feeble grounds that the text is readily available online. (Given the choice, I’d have thought that many readers might have preferred the read-once Note on the Text to be mostly an online supplement, and the might-want-to-consult-often German original to be included.) There’s an Appendix giving the top-level numbered propositions again. And then twenty pages of Explanatory Notes, which often  comment on the original German, making the absence of the original again an oddity. And finally there is a two-part Glossary, first German-English and then English-German.

How useful is the Introduction? Much more importantly, how good is the Translation?

On the first, I’d say the Introduction is not particularly good or clear. Perhaps it tries to do too much in too short a space. The student new to the Tractatus would do much better to read the (albeit rather longer but wonderfully clear) chapters in Antony Kenny’s still remarkable 1973 Wittgenstein or perhaps the main chapters of Roger White’s still short 2006 Wittgenstein’s Tractatus Logico-Philosophicus. 

As for the Translation, I’m in no position to really judge. But, almost at random, here’s the Pears/McGuinness rendering of 4.026:

The meanings of simple signs (words) must be explained to us if we are to understand them. With propositions, however, we make ourselves understood.

Beaney has

The meanings of simple signs (words) must be explained to us for us to understand them. With propositions, however, we communicate.

The repeated “us” in the first sentence is unnecessarily ugly. And Beaney adds a note on the second sentence “lit. we make ourselves understood, connecting with the use of ‘verstehen’ (‘understand’) in the previous sentence” which makes the departure from Pears/McGuinness seem a bit puzzling. Is Beaney’s version an improvement?

Again, here’s the Pears/McGuinness translation of 4.041

This mathematical multiplicity, of course, cannot itself be the subject of depiction. One cannot get away from it when depicting.

And Beaney:

This mathematical multiplicity, of course, cannot in it turn be depicted. One cannot get outside it in depiction.

Is the second sentence even English?

And so it goes. I’m not immediately bowled over. However, it is evident that a great deal of thought and widely-sought advice has gone into shaping the translation, and many of the notes on translation look well-judged. Beaney’s final explanatory note is a rather engaging two-page essay on why he has, for example, rendered the famed last sentence of the Tractatus as “Of what one cannot speak, about that one must be silent.”

As to content of the Tractatus itself, about that I must indeed be silent!

The Study Guide, corrected reprint

There is now a corrected update of the Beginning Mathematical Logic Study Guide. The list of known typos for the 2022 printing was getting embarrassingly long; so I’ve taken the opportunity to correct these plus the one thinko noted on the corrections page. Otherwise little has changed, apart from some minor rephrasing here and there. There’s certainly no need to rush to order a new copy if you already have one!

The corrected PDF is now available for download. The print-on-demand version should update to match in due course (depending on whether your local Amazon has any small stocks of the current printing to clear). So if you were thinking of buying yourself a copy as a New Year’s treat, you might want to hold off for a week or two.

I plan to intermittently work on a revised second edition over the coming year, improving some of the topic overviews, and — this will take some time and effort! — revisiting some of the sets of recommendations. Still, in reading through parts of the Guide while preparing the corrected reprint, I mostly still thought reasonably well of it in its current state; and so I hope this very minor update will serve for the moment.

Scroll to Top